# What is the projection of < -3, 7, 7> onto < -3 , 4, -5>?

Oct 19, 2017

The projection is $= \frac{1}{25} < - 3 , 4 , - 5 >$

#### Explanation:

The vectors are

$\vec{a} = < - 3 , 7 , 7 >$

$\vec{b} = < - 3 , 4 , - 5 >$

The projection of $\vec{a}$ onto $\vec{b}$ is

$p r o {j}_{\vec{b}} \vec{a} = \frac{\vec{a} . \vec{b}}{| | \vec{b} | {|}^{2}} \vec{b}$

$= \frac{< - 3 , 7 , 7 > . < - 3 , 4 , - 5 >}{| | < - 3 , 4 , - 5 > | {|}^{2}} < - 3 , 4 , - 5 >$

$= \frac{{\left(- 3\right)}^{2} + \left(7.4\right) + \left(7. - 5\right)}{{\left(- 3\right)}^{2} + \left(4.4\right) + \left(- 5. - 5\right)} < - 3 , 4 , - 5 >$

$= \left(\frac{2}{50}\right) < - 3 , 4 , - 5 >$

$= \frac{1}{25} < - 3 , 4 , - 5 >$